# VERBAL JUSTIFICATION—IS IT A PROOF? SECONDARY SCHOOL TEACHERS’ PERCEPTIONS

- 274 Downloads
- 8 Citations

## ABSTRACT

According to reform documents, teachers are expected to teach proofs and proving in school mathematics. Research results indicate that high school students prefer verbal proofs to other formats. We found it interesting and important to examine the position of secondary school teachers with regard to verbal proofs. Fifty high school teachers were asked to prove various elementary number theory statements, to write correct and incorrect proofs that students may use, and to evaluate given justifications to statements from elementary number theory. While all the participants provided correct proofs to the statements, our findings indicate that teachers are not aware of students’ preference for verbal justifications. Also, about half of the teachers rejected correct verbal justifications. They claimed that these justifications lacked generality and are mere examples.

## KEY WORDS

elementary number theory teachers’ knowledge about proofs verbal proof## Preview

Unable to display preview. Download preview PDF.

## References

- Aharon, Y. (2003).
*High school students’ conceptions of proofs and refutations*. Unpublished M.A. thesis. Tel-Aviv, Israel: Tel Aviv University (in Hebrew).Google Scholar - Australian Education Council. (1991).
*A national statement on mathematics for Australian schools*. Melbourne, Victoria, Australia: Curriculum Corporation.Google Scholar - Ben Yaacov, E. (2004).
*The gap between teachers’ and students’ perceptions regarding proofs*. Unpublished M.A. thesis. Tel-Aviv, Israel: Tel Aviv University (in Hebrew).Google Scholar - Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2002). Proving or refuting arithmetic claims: The case of elementary school teachers. In A. D. Cockburn & E. Nardi (Eds.),
*Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education, Vol. 1*(pp. 57–64). Norwich, UK: University of Norwich.Google Scholar - Caspi, S. (2005).
*Students’ validations and refutations of different types of arithmetical statements: The case of inductive approach*. Unpublished M.A. thesis. Tel-Aviv, Israel: Tel Aviv University (in Hebrew).Google Scholar - Dreyfus, T. (2000). Some views on proofs by teachers and mathematicians. In A. Gagatsis (Ed.),
*Proceedings of the 2nd Mediterranean conference on mathematics education, Vol. I*(pp. 11–25). Nikosia, Cyprus: The University of Cyprus.Google Scholar - Edwards, L. D. (1999). Odd and even: Mathematical reasoning processes and informal proofs among high school students.
*Journal of Mathematical Behavior, 17*, 498–504.Google Scholar - Healy, L., & Hoyles, C. (1998).
*Justifying and proving in school mathematics*. University of London, Institute of Education: Technical Report.Google Scholar - Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra.
*Journal for Research in Mathematics Education, 31*, 396–428.CrossRefGoogle Scholar - Hill, H., Ball, L. D., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students.
*Journal for Research in Mathematics Education, 39*(4), 372–400.Google Scholar - Israeli Ministry of Education. (1994).
*Tomorrow 98*. Jerusalem, Israel: Ministry of Education. in Hebrew.Google Scholar - Messinger, R. (2006).
*Students’ validations and refutations statements of type “For all” and “Exist”: The case of deductive approach*. Unpublished M.A. thesis. Tel-Aviv, Israel: Tel Aviv University (in Hebrew).Google Scholar - National Council of Teachers of Mathematics (2009).
*Focus in high school mathematics: Reasoning and sense making.*Reston, VA: National Council of Teachers of Mathematics. Retrieved October 10, 2009, from http://www.nctm.org/ - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Schoenfeld, A. H. (1994). What do we know about mathematics curricula?
*Journal of Mathematical Behavior, 13*, 55–80.CrossRefGoogle Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. New York: Academic Press.Google Scholar - Shulman, L. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher, 15*(2), 4–14.Google Scholar - Stylianides, A. (2007). Proof and proving in school mathematics.
*Journal for Research in Mathematics Education, 38*, 289–321.Google Scholar - Tabach, M., Barkai, R., Tirosh, D., Tsamir, P., & Dreyfus, T. (2009).
*Secondary school teachers’ awareness of numerical examples as proof*. Technical report, available from the authors.Google Scholar - Tirosh, C. (2002).
*The ability of prospective teachers to prove or to refute arithmetic statements.*Unpublished Doctoral dissertation. Jerusalem, Israel: The Hebrew University (in Hebrew).Google Scholar - Tsamir, P., Tirosh, D., Dreyfus, T., Barkai, R., & Tabach, M. (2008). Inservice teachers’ judgment of proofs in ENT. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.),
*Proceedings of the 32nd conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 345–352). Morélia, México: PME.Google Scholar - Tsamir, P., Tirosh, D., Dreyfus, T., Barkai, R., & Tabach, M. (2009). Should proof be minimal? Ms T’s evaluation of secondary school students’ proofs.
*Journal of Mathematical Behavior, 28*, 58–67.CrossRefGoogle Scholar